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Lecture 2. Lambda Abstraction, NP Semantics, and a Fragment of English

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Lecture 2. Lambda Abstraction, NP Semantics, and a Fragment of English Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.1 B. Partee, MGU, February 22, 2005 p.2 Lecture 2. Lambda abstraction, NP semantics, and a Fragment of English (b) NP 1. Lexical and Structural Ambiguity..............................................................................................................................1 | 2. Lambdas ....................................................................................................................................................................3 CNP 2.1. A first-order part of the lambda-calculus............................................................................................................3 3 2.2. The typed lambda calculus. ................................................................................................................................4 ADJ CNP 3. Montague’s semantics for Noun Phrases...................................................................................................................5 | 3.1. Semantics via direct model-theoretic interpretation of English..........................................................................5 9 3.2. Semantics via translation from English into a logical language. ........................................................................5 old CNP and CNP 4. English Fragment 1...................................................................................................................................................5 | | 4.0 Introduction ........................................................................................................................................................5 men women 4.1. Syntactic categories and their semantic types.....................................................................................................6 4.2. Syntactic Rules and Semantic Rules...................................................................................................................6 (2) Every student read a book. (Quantifier scope ambiguity) 4.2.1. Basic syntactic rules.....................................................................................................................................7 4.2.2. Semantic interpretation of the basic rules...................................................................................................7 Just one (surface) syntactic structure: 4.2.3. Rules of Relative clauses, Quantification, Phrasal Negation. See Section 6. ..............................................8 4.2.4. Type multiplicity and type shifting..............................................................................................................8 S 4.3. Lexicon. ..............................................................................................................................................................9 3 5. Examples .................................................................................................................................................................11 NP VP 6. Rules of Relative clause formation, Quantifying In, Phrasal Negation. ..................................................................11 6.1. (Restrictive) Relative clause formation. ...........................................................................................................11 3 3 6.2. Quantifying In...................................................................................................................................................12 DET CNP V NP 6.3. Conjunction. .....................................................................................................................................................13 | | | 3 6.4. Phrasal and lexical negation. ............................................................................................................................14 every student read DET CNP Appendix: Montague’s intensionsal logic, with lambdas and types. .........................................................................15 | | A.1 Introduction.......................................................................................................................................................15 a book A.2. Intensional Logic (IL)......................................................................................................................................15 A.2.1. Types and model structures. .....................................................................................................................15 Predicate logic representations of the two readings: A.2.2. Atomic expressions (“lexicon”), notation, and interpretation...................................................................16 (i) ∀x ( Student (x) → y ( Book (y) & Read (x,y)) A.2.3. Syntactic rules and their model-theoretic semantic interpretation ............................................................17 › REFERENCES. ...........................................................................................................................................................18 (ii) ∃y ( Book (y) & ∀x ( Student (x) → Read (x,y)) HOMEWORK #1, Due March 8. ................................................................................................................................19 HELP FOR HOMEWORK #1.....................................................................................................................................21 Compositional interpretation of the English sentence: ??. More below. NOTE: I know this is too much for one lecture. But it is useful to have all of this in one handout. We will spend this time and part of next time on this, with more next time about the semantics of noun phrases as Generalized The difficulty for compositionality if we try to use predicate calculus to represent “logical Quantifiers. form”: What is the interpretation of “every student”? There is no appropriate syntactic category or semantic type in predicate logic. Inadequacy of 1st-order predicate logic for 1. Lexical and Structural Ambiguity representing the semantic structure of natural language. We can solve this problem when we Lexical ambiguity: bank1, bank2 : both CN (common noun), homonyms; have the lambda-calculus and a richer type theory. open1 (ADJ), open2 (IV) (intransitive verb), open3 (TV) (transitive verb). Categories of PC: Categories of NL: Structural ambiguity. Compositionality requires a “disambiguated language” (a “language Formula - Sentence without ambiguity”). So we interpret expressions with syntactic structure, not just strings. Predicate - Verb, Common Noun, Adjective Term (1) old men and women. Two meanings, two structures. “old” applies only to “men”, or to Constant - Proper Noun “men and women”. Variable - Pronoun (he, she, it) ========== (a) NP (no more) - Verb Phrase, Noun Phrase, Common Noun Phrase, Adjective 9 NP and NP Phrase, Determiner, Preposition, Prepositional Phrase, Adverb, | | CNP CNP 3 | ADJ CNP women | | old men MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.3 B. Partee, MGU, February 22, 2005 p.4 2. Lambdas Rule for combining CNP and REL: λy[CNP’(y) & REL’(y)] (combining “translations”) 2.1. A first-order part of the lambda-calculus. Compositional translation of the syntactic structure above into λ-calculus: (read bottom-to- To begin looking at the lambda calculus, we will start with just a “first-order” part of top) it, as if we were just adding a bit of the lambda calculus to the predicate calculus rules from λy[man(y) & λz[love (z,m)] (y)] Lecture 1. Then in section 2.2. we will look at the fully typed lambda calculus as given in 3 Montague’s Intensional Logic Rule 7 in the Appendix. man λz[love (z,m)] | | Lambda-abstraction rule, first version. man love (z, m) λ-abstraction applies to formulas to make predicates. This extends PC in a way that allows us to represent more complex Common Noun Phrases, Adjective Phrases, some Verb By λ-conversion (see below), the top line is equivalent to: λy[man(y) & love (y,m)] Phrases. For some other categories we will need the full version of the λ-abstraction rule. R9: If ϕ ∈ Form and v is a variable, then λv[ϕ] ∈ Pred-1. 2.2. The typed lambda calculus. S9: ›λv[ϕ]M, g is the set S of all d 0 D such that || ϕ ||M, g [d/v] = 1. Examples. For all examples, assume that we start with an assignment g such that g(v) = John The full version of the typed lambda calculus fits into Montague’s intensional logic with its for all v. In most of the examples below, the choice of initial assignment makes no type theory; see the Appendix for a complete statement of Montague’s intensional logic. The difference. And assume that I(b) = Bill, I(m) = Mary. parts we will use the most will be the type theory, the lambda calculus (Rule 7), and the rule of “functional application” (Rule 6). Montague’s intensional logic includes the predicate M, g (i) ›|| λx[run(x)] || = the set of all individuals that run. calculus as a subpart (see Rule 2), but not restricted to first-order: we can quantify over variables of any type. (ii) ›|| λx[love(x, b)] ||M, g = the set of all individuals that love || b ||M, g[d/x], i.e. I(b), i.e. Bill. M, g M, g[d/x] (iii) ›|| λx[love(x, y)] || = the set of all individuals that love || y || , i.e. g(y), i.e. John. Lambda-abstraction, full version. (iv) ›|| λx[fish (x) & love(x, b)] ||M, g = the set of all fish that love Bill. In general: λ-expressions denote functions. λv[α] denotes a function whose argument
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